There are three places where a super-linear dependence on \(n\)  appears:
the application of the irrelevant vertex technique takes \(2^{O(k)}n^2\)  time,
the computation of the backbone tree takes \(O(n^2)\)  time, and the algorithm of [1]} takes \(O(n^6)\)  time.
Although a fine-grained analysis of the algorithm of [1]} gives the bound of
\(O(\chi ^2 + n\cdot \eta \cdot \chi \cdot |\Pi |)\) , the values of \(\chi , \eta \)  and \(|\Pi |\)  are too large in the case of the weak \(T\) -linkages \(\mathcal {W}\) 
constructed from [3]}.
This is because the number of edges of \(R\)  is \(\Theta (n)\) , and the maximum radial distance from the outer face to
the vertices of \(R\)  is \(\Theta (n)\)  in the worst case.
